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A diffusion equation for linear fractional stable motion, apparent multifractality \& applications to space physics
by: N.W. Watkins and D. Credgington and R. Sanchez and S.C. Chapman
In: Abstract Book of the XXIII IUPAP International Conference on Statistical PhysicsGenova, Italy:
, 9-13 July
(2007)
.
Resources (URL, PDF, PS...)
Abstract
In the 1960s Mandelbrot developed the use of fractals to describe
how the shape of many aspects of the natural world departs from the
Euclidean. In particular he proposed two kinds of fractal model to
capture the way in which natural data is often persistent in time
his Joseph effect, common in hydrology and exemplified by
fractional Brownian motion and or prone to heavy tailed jumps the Noah
effect, typical of economic index time series, for
which he gave L\'evy flights as an exemplar. Both effects are now
well demonstrated in proxies both for the Earth's auroral electric
currents and for the turbulent solar wind which is their ultimate energy
source. Modelling, however, has usually emphasised one of the
Noah and Joseph parameters the tail exponent $\mu$ and one derived
from the temporal behaviour such as power spectral $\beta$ at the other's
expense. This poster will first describe recent work 1 in which we applied a simple self-affine stable model-linear fractional stable motion, LFSM, which unifies both effects-to give insight into space physics data. I will show how we have resolved some contradictions seen in earlier work, where purely Joseph or Noah descriptions had been sought. Such hybrid Noah-Joseph ambivalent 2 behaviour is highly
topical in physics. It is typically studied in the paradigm of the
continuous time random walk CTRW rather than LFSM. Intriguingly the
self-similarity exponent extracted from the CTRW differs from that
seen in LFSM, being a ratio of $\mu$ and a temporal exponent rather
than an additive function. The poster will elucidate the physical
differences between these two pictures with reference to a newly-derived diffusion equation for LFSM, which replaces the second order spatial derivative in the equation of fBm 3 with a fractional derivative of order $\mu$.
I will also show work in progress using an LFSM generator and simple
analytic scaling arguments to study the problem of the area between a
fractional L\'evy curve and a threshold-related both to Bernoulli excursions
and to the burst size measure introduced by Takalo and Consolini into solar-terrestrial physics and further studied by Freeman et al 4,5.
Finally I will discuss how LFSM gives the appearance of multi-affine scaling without having an underlying turbulent cascade or other multiplicative process. The importance of this property for the interpretation of natural time series will be discussed.
1 Watkins et al, Space Sci. Rev. 121, 271, 2005.\\
2 Brockmann et al, Nature 439, 462, 2006.\\
3 Wang and Lung, Phys. Lett. A 151, 119, 1990.\\
4 Freeman et al, Geophys. Res. Lett. 27, 1367, 2000.\\
5 Freeman et al, Phys. Rev. E 62, 8794, 2000.


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